Norm of Gaussian integers in arithmetical progressions and narrow sectors
We proved the equidistribution of the Gaussian integer numbers in narrow sectors of the circle of radius x¹/² , x → ∞, with the norms belonging to arithmetic progression N(α) ≡ ℓ (mod q) with the common difference of an arithmetic progression q, q ≪ x²/³⁻ᵋ.
Saved in:
| Published in: | Algebra and Discrete Mathematics |
|---|---|
| Date: | 2020 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2020
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/188520 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Norm of Gaussian integers in arithmetical progressions and narrow sectors / S. Varbanets, Y. Vorobyov // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 2. — С. 259–270. — Бібліогр.: 4 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860203499678072832 |
|---|---|
| author | Varbanets, S. Vorobyov, Y. |
| author_facet | Varbanets, S. Vorobyov, Y. |
| citation_txt | Norm of Gaussian integers in arithmetical progressions and narrow sectors / S. Varbanets, Y. Vorobyov // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 2. — С. 259–270. — Бібліогр.: 4 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | We proved the equidistribution of the Gaussian integer numbers in narrow sectors of the circle of radius x¹/² , x → ∞, with the norms belonging to arithmetic progression N(α) ≡ ℓ (mod q) with the common difference of an arithmetic progression q, q ≪ x²/³⁻ᵋ.
|
| first_indexed | 2025-12-07T18:11:32Z |
| format | Article |
| fulltext |
“adm-n2” — 2020/7/8 — 9:42 — page 259 — #119
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 29 (2020). Number 2, pp. 259–270
DOI:10.12958/adm1529
Norm of Gaussian integers in arithmetical
progressions and narrow sectors
S. Varbanets and Y. Vorobyov
Communicated by Yu. A. Drozd
Abstract. We proved the equidistribution of the Gaussian
integer numbers in narrow sectors of the circle of radius x
1
2 , x → ∞,
with the norms belonging to arithmetic progression N(α) ≡ ℓ
(mod q) with the common difference of an arithmetic progression q,
q ≪ x
2
3
−ε.
Introduction
For the classical arithmetic functions τ(n) (the number of divisors for
the positive integer n) and r(n) (the number of representations for the
positive integer n as sum of two squares of integers) there were obtained
the asymptotic formulas of the sums
∑
n≡ℓ (mod q)
n6x
τ(n) and
∑
n≡ℓ (mod q)
n6x
r(n),
where q grows together with x and they are nontrivial for q ≪ x
2
3
−ε.
For the function τ(n) K. Liu, I. Shparlinskii and T. Zhang ([2]) obtained
the extended region of non-triviality.
In the present paper we investigate the distribution of points from
complex plane C =
{
x+ iy
∣
∣x, y ∈ R
}
, ϕ1 < arg (x+ iy) 6 ϕ2, ϕ2−ϕ1 <
2010 MSC: 11L07, 11T23.
Key words and phrases: Gaussian integers, norm groups, Hecke Z-function,
functional equation.
https://doi.org/10.12958/adm1529
“adm-n2” — 2020/7/8 — 9:42 — page 260 — #120
260 Norm of Gaussian integers
π
2 , x2 + y2 ≡ ℓ (mod q), x2 + y2 6 N . Using the property of Hecke Z-
function of the quadratic field Q(i) and the estimates of special exponential
sums, we obtain a non-trivial asymptotic formula for the number of integer
points under the circle’s sectorial region in arithmetic progression with
the growing difference progression.
Throughout this paper we use the following notations.
• p denotes a prime number in Z;
• the Latin letters a, b, k, m, n, ℓ be the positive integers;
• ℜz denotes the real part of z and ℑz be the imaginary part of z;
• through Z we denote the ring of integers;
• G = Z[i] denotes the ring of Gaussian integers a + bi, a, b ∈ Z,
i2 = −1;
• Gγ (respectively, G∗
γ) be the ring of residue classes modulo γ (re-
spectively, the multiplicative group of inversive element in Gγ);
• N(ω) is the norm of ω ∈ G, N(ω) = |ω|2;
• Sp(ω) is the trace of ω from Q(i) to Q, Sp(ω) = 2ℜω;
• symbols “≪” and “O” are equivalent;
• s = σ + it ∈ C, ℜs = σ, ℑs = t;
• χq denotes the Dirichlet character modulo q over Z
• (a, q) = gcd (a, q) in Z;
• (α, ω) = gcd (α, ω) in G;
1. Auxiliary results
Let δ1, δ2 ∈ Q(i) and s = σ+ it. For the rational integer number m let
us define the function sized by absolutely convergent series into semiplane
ℜs > 1:
Zm(s, ; δ1, δ2) :=
∑
ω∈G
e4mi argω+δ1
N(ω + δ1)s
eπiSp(δ2·ω).
It is obvious that with m = 0 we get the Epstein zeta-function. With
δ1, δ2 ∈ Qi we get the Hecke Z-function over the imaginary quadratic field
Q(i).
Let p > 2 be a prime rational number, n ∈ N. Denote
Epn :=
{
α ∈ Gpn
∣
∣N(α) ≡ ±1 (mod pn)
}
. (1)
It is also obvious that En is the subgroup of multiplicative group of
residue classes modulo pn over the ring Gpn .
We call Epn the norm group in G∗
pn .
“adm-n2” — 2020/7/8 — 9:42 — page 261 — #121
S. Varbanets, Ya. Vorobyov 261
Lemma 1. Let p ≡ 3 (mod 4) and En be the norm group in Gpn. Then
En is the cyclic group, |En| = 2(p+1)pn−1, and let u+ iv be a generative
element of En. Then exist x0, y0 ∈ Z∗
pn such that
(u+ iv)2(p+1) ≡ 1 + p2x0 + ipy0,
2x0 + y20 ≡ −2p2x20 (mod p3).
Moreover, we have modulo pn for any t = 4, 5, . . . , pn−1 − 1,
ℜ
(
(u+ iv)2(p+1)t
)
= A0 +A1t+A2t
2 + · · ·
ℑ
(
(u+ iv)2(p+1)t
)
= B0 +B1t+B2t
2 + · · · ,
where
A0 ≡ 1 (mod p4), B0 ≡ 0 (mod p4),
A1 ≡ p2x0 +
1
2
p2y20 ≡ −
5
2
x20p
4 (mod p5),
B1 ≡ py0(1− p2x0) (mod p4),
A2 ≡ −
5
2
x20p
2 (mod p5), B2 ≡
5
3
p3x0y0 (mod p4),
Aj ≡ Bj ≡ 0 (mod p3), j = 3, 4, . . .
(In greater details see [3])
Denote
(u+ iv)k = u(k) + iv(k), 0 6 k 6 2p+ 1,
(u+ iv)2(p+1)t+k ≡
n−1
∑
j=0
(Aj(k) + iBj(k))t
k (mod pn).
It is obvious that
Aj(k) = Aju(k)−Bjv(k), Bj(k) = Ajv(k) +Bju(k).
Thus from Lemma 1 we infer
Corollary. For k = 0, 1, . . . , 2p+ 1 we have
u(0) = 1, v(0) = 0, (u(p+ 1), p) = 1, p||v(p+ 1);
(u(k), p) = (v(k), p) = 1 for k 6≡ 0 (mod
p+ 1
2
);
u(k) ≡ 0 (mod p), (v(k), p) = 1 if k =
p+ 1
2
or
3p+ 1
2
;
u(k) ≡ u(−k), v(k) ≡ −v(−k).
“adm-n2” — 2020/7/8 — 9:42 — page 262 — #122
262 Norm of Gaussian integers
Hence, for k 6≡ 0 (mod p+1
2 ) we have
A0(k) ≡ u(k) (mod p), B0(k) ≡ v(k) (mod p),
A1(k) ≡ −py0v(k), B1(k) ≡ py0u(k) (mod p2)
A2(k) ≡ −
5
2
x20p
2u(k), B2(k) ≡ −
5
2
x20p
2v(k) (mod p4).
(2)
For k = p+1
2 or 3p+1
2 we obtain
p||A1(k), p2||B1(k), p2||A2(k), B2(k) ≡ 0 (mod p3). (3)
Moreover,
A1(0) ≡ −
5
2
x20p
4 (mod p5), B1(0) ≡ 0 (mod p4),
A2(0) ≡ −
5
2
x20p
2 (mod p5), B2(0) ≡ 0 (mod p3), p2||A1(p+ 1),
p||B1(p+ 1), p2||A2(p+ 1), B2(p+ 1) ≡ 0 (mod p3).
(4)
At last for all k = 0, 1, . . . , 2p+ 1
Aj(k) ≡ Bj(k) ≡ 0 (mod p3), j = 3, 4, . . . .
Lemma 2. Let q = pℓ with ℓ > 1, g(y) is the polynomial in form
g(y) = A1y + pA2y
2 + pλ3A3y
3 + · · ·+ pλkAky
k, k > 3,
with Aj ∈ Z, (Aj , p) = 1, j = 3, . . . , k, 2 6 λ3 6 λ4 6 · · · 6 λk. Then we
have
Sq :=
q−1
∑
y=1
e
2πi
g(y)
pℓ = p[
ℓ
2 ]
∑
y∈Z
p[ℓ/2]
g′(y)≡0 (mod p[ℓ/2])
Bq(y), (5)
where
Bq(y) =
0 if (A1, p) = 1,
1 if ℓ ≡ 0 (mod 2),
A1 ≡ 0 (mod p),
∑p−1
z=0 e
2πi
((
A1
p +2A2
)
z+2z2
)
p if ℓ ≡ 1 (mod 2),
A1 ≡ 0 (mod p).
Proof. The proof of this assertion repeats the proofs of Lemmas 12.3 and
12.4 in [1].
“adm-n2” — 2020/7/8 — 9:42 — page 263 — #123
S. Varbanets, Ya. Vorobyov 263
For p ≡ 1 (mod 4) or p = 2 the norm groups are not the cyclic groups.
We shall use the description of the solutions x2 + y2 ≡ 1 (mod pn) for
these cases.
Lemma 3. Let (x, y) is a solution of the congruence x2+y2 ≡ 1 (mod pℓ),
p > 2 is a prime number. Then all solutions with (x0, p) = 1 are described
in the following manner
x = x(0)f(y0, t), y = y0 + pt, t = 0, 1, . . . , pℓ−1 − 1, (6)
where x(0) runs all solutions of the congruence
x2 ≡ 1− y20 (mod pn),
y0 runs all solutions of the congruence
x20 + y20 ≡ 1 (mod p)
with x0 6≡ 0 (mod p), and
f(y0, t) = 1 + p
y0
y20 − 1
t+ p2
1− y0
y20 − 1
t2 + pλ3X3(y0)t
3 + · · ·+ pλsXs(y0)t
s,
under conditions (Xj(y0), p) = 1, λj > 3, s 6
[
ℓp−1
p−2
]
.
For the solutions of the congruence x2 + y2 ≡ 1 (mod pℓ) with x0 ≡ 0
(mod p) we have
x = pt, y ≡ ±
(
1−
1
2
p2t2
)
(mod p4). (7)
(Here, the multiplicative inverse for 2 and y20 − 1 is considered modulo pn).
Lemma 3′. Let s =
[
ℓ−1
2
]
. There exists the polynomial
f(t) = 1 + 2λ1A− 1t2 + · · ·+ 2λsAst
2s,
with Aj ≡ 1 (mod 2), λj > 2j + 1, j = 1, . . . , s, such that all solutions of
the congruence x2 + y2 ≡ 1 (mod pℓ) can be written as
x = 4t, y = ±f(t) or x = 4t, y = ±(2ℓ−1 − 1)f(t),
t = 0, 1, . . . , 2ℓ−2 − 1.
(8)
“adm-n2” — 2020/7/8 — 9:42 — page 264 — #124
264 Norm of Gaussian integers
Lemma 4. Let us I(ℓ, q) be the number of solutions of the congruence
u2 + v2 ≡ a (mod q), (a, q) =
∏
p|q
pt0 .
Then we have
I(a, q)
= c(a, q)q
∏
pt||q
(
1−
χ4(p
t0+1)
p
(
1− χ4(p
t−t0)
)
+
(
1−
1
p
)
t−1
∑
b=t−t0
χ4(p
t−b)
)
,
where
c(a, q) =
1 if (q, 2) = 1,
1 if 2||q,
1 if q ≡ 0 (mod 4), t0 > t− 2,
2 if q ≡ 0 (mod 4), t0 < t− 2 and a
2t0
≡ 1 (mod 4),
0 if q ≡ 0 (mod 4), t0 6 t− 2 and a
2t0
≡ 3 (mod 4)
This lemma follows from the equation
I(a, pt) =
∑
u,v∈Zpt
1
pt
∑
z∈Z∗
pt
e
2πi
z(y2+v2−ℓ)
pt
and the values of the Gaussian sums
∑
x∈Zpt
e
2πi zx
2
pt .
Similarly, we obtain the description of the solutions of the congruence
x2 + y2 ≡ −1 (mod pℓ), p ≡ 1 (mod 4). Indeed, let c0 be the solution of
the congruence x2 ≡ −1 (mod pℓ). Then
x = c0x(0)f1(y0, t), y = y0 + pt, t = 0, 1, . . . , pℓ−1 − 1,
where f1(y0, t) is as f(y0, t).
2. The main results
We consider the generalized Hecke Z-function of quadratic field Q(i)
Zm(s; δ1, δ2) :=
∑
ω∈G
ω 6=δ1
e4mi arg(ω+δ1)
N(ω + δ1)
eπiSp(ωδ2), (ℜs > 1),
“adm-n2” — 2020/7/8 — 9:42 — page 265 — #125
S. Varbanets, Ya. Vorobyov 265
where δ1, δ2 ∈ Q(i), m ∈ Z. This function satisfies the functional equation
π−1Γ(2|m|+ s)Zm(s; δ1, δ2)
= π−(1−s)Γ(2|m|+ 1− s)Z−m(1− s;−δ2, δ1)e
πiSp(δ1δ2).
(9)
The function Zm(s; δ1, δ2) is an entire function except the case m = 0
and the Gaussian integer δ2 when Zm(s; δ1, δ2) is holomorphic for all
complex s exclusive s = 1 where it has a simple pole with residue π.
We define the multiplicative character modulo q over G∗
pℓ
as
χ(ω) = χpℓ(N(ω)),
where χpℓ is the character modulo pℓ in Z∗
pℓ
.
Let Ξm(ω) := e4mi argωχ(ω) = e4mi argωχpℓ(N(ω)). Then from (9) we
have for Z(s; Ξm) :=
∑
ω
Ξm(ω)
N(ω)s the following functional equation
Z(s; Ξm) = κ(Ξm)Ψ(s,Ξm)Z(1− s,Ξm), (10)
where
κ(Ξm) = (N(pℓ))−
1
2
∑
τ∈G
pℓ
χ(N(τ))e
Sp τ
pℓ ,
Ψ(s,Ξm) =
(
1
π
N(pℓ)
1
2
)1−2s Γ(2|m|+ 1− s)
Γ(2|m|+ s)
.
(11)
Denote
rm(n) =
∑
u,v∈Z
u2+v2=n
e4mi arg (u+iv).
From this we have
∑
n6x
rm(n)χpℓ(n) =
∑
u,v∈Z
u2+v2=n6x
e4mi arg (u+iv)χpℓ(n).
Therefore,
Fm(s) =
∞
∑
n=1
rm(n)
ns
=
∑
χq
χq(a) · Z(s; Ξm).
“adm-n2” — 2020/7/8 — 9:42 — page 266 — #126
266 Norm of Gaussian integers
We get by the Perron’s formula on an arithmetic progression with c > 1,
T > 1, (a, pℓ) = 1, 0 < ε < 1
2 the following equality
∑
n≡a (mod pℓ)
n6x
rm(n) =
1
2πi
∫ c+iT
c−iT
Fm(s)
xs
s
ds+O
(
xc
Tpℓ(c− 1)
)
+O (xε)
= res
s=0,1
(
Fm(s)
xs
s
)
+
1
2πi
∫ −ε+iT
−ε−iT
Fm(s)
xs
s
ds+ max
−ε6ℜs6c
∣
∣
∣
∣
1
s
Fm(s)xs
∣
∣
∣
∣
+O
(
xc
Tpℓ(c− 1)
)
+O (xε) ,
(12)
where ε is a positive arbitrary small number.
From the functional equation for Z(s,Ξ), summing all over character
χpℓ , we have for ℜs < 0
Fm(s) = π−1+2sΓ(2|m|+ 1− s)
Γ(2|m|+ s)
×
∑
ω∈G
(ω,pℓ)=1
e−4mi argω
N(ω)1−s
∑
τ∈G∗
pℓ
N(τ)≡aN(ω) (mod pℓ)
e
Sp(τ)
pℓ .
Consider the sum
∑
0
:=
∑
τ∈G∗
pℓ
N(τ)≡n (mod pℓ)
(n,pℓ)=1
e
πiSp( τ
pℓ
)
.
For p ≡ 3 (mod 4), we apply the representation of elements from the
norm group Epℓ . Lemma 1 and its Corollary give
∑
0
=
2p+1
∑
k=0
e
2πi
A′
0(k)
pℓ
pℓ−1−1
∑
t=0
e
2πi
A′
1(k)t+A′
2(k)t
2+···
pℓ
= e
2πi
A′
0(0)
pℓ
pℓ−1−1
∑
t=0
e
2πi
A′
1(0)t+A′
2(0)t
2+···
pℓ
+ e
2πi
A′
0(p+1)
pℓ
pℓ−1−1
∑
t=0
e
2πi
A′
1(p+1)t+A′
2(p+1)t2+···
pℓ ,
where A′
j(0) and A′
j(p+1) differ from Aj(j) and A− j(p+1) only by the
multiplier N(ω)a.
“adm-n2” — 2020/7/8 — 9:42 — page 267 — #127
S. Varbanets, Ya. Vorobyov 267
Now Lemma 3 gives
E0 = p
ℓ
2
(
e
2πi
A′
0(0)
pℓ + e
2πi
A′
0(p+1)
pℓ
)
×
{
1 if ℓ ≡ 0 (mod 2),
e
−
2πiA′
1(2A
′
2)
−1
p if ℓ ≡ 1 (mod 2).
(13)
If p ≡ 1 (mod 4) or p = 2 we use Lemma 1 and then obtain E0 =
O
(
pℓ
1
2
)
with an absolute constant in the symbol "O".
Now we able to prove the main theorems.
Let us denote through A(x;ϕ1, ϕ2; a, p
ℓ) the number of points (u, v)
in the circle (u2 + v2) 6 x under conditions
u, v ∈ Z, ϕ1 < arg (u+ iv) 6 ϕ2,
u2 + v2 ≡ a (mod pℓ), (a, pℓ) = 1.
(14)
Theorem 1. For x → ∞ the following estimate
∑
n≡a (mod pℓ)
n6x
rm(n) = ε
πx
pℓ
k0
(
1−
χ4(p)
p
)
+O
(
x
1
2
+ε
p
ℓ
4
M1+ε
)
+O
(
p
ℓ
2M1+ε
)
,
(15)
holds, where εm = 0 if m 6= 0, ε0 = 1, k0 = 1 if p > 2, or k = 2 if p = 2,
ℓ > 3; M = |m|+ 3, ε > 0 is an arbitrary small number; constants in the
symbols can depend only on ε.
Proof. The function Fm(s) has a pole in s = 1 only if m = 0:
res
s=1
F0(s) =
πx
pℓ
k0
(
1−
χ4(p)
p
)
.
The estimate for Fm(0) is easy proving by the Phragmen-Lindelöff principle
and the estimates of Zm(s) on the bounds of stripe −ε 6 ℜs 6 1 + ε.
Therefore, we have
res
s=0
Fm(s) ≪ p
ℓ
2 (|m|+ 3) log (|m|+ 3).
“adm-n2” — 2020/7/8 — 9:42 — page 268 — #128
268 Norm of Gaussian integers
Hence,
∑
n≡a (mod pℓ)
n6x
rm(n) = εm
πx
pℓ
∑
u,v∈Z
pℓ
u2+v2≡a (mod pℓ)
1 +O
(
p
ℓ
2 (|m|+ 3) log (|m|+ 3)
)
+
+
1
2πi
∫ −ε+iT
−ε−iT
Fm(s)
xs
s
ds+O
(
xc
Tpℓ(c− 1)
+ xε
)
.
(16)
Note that
εm
πx
pℓ
∑
u,v∈Z
pℓ
u2+v2≡a (mod pℓ)
1 = εm
πx
pℓ
k0
(
1−
χ4(p)
p
)
,
where
Fm(s) = π−1+2sΓ(2|m|+ 1− s)
Γ(2|m|+ s)
×
∑
ω∈G
(ω,pℓ)=1
e−4mi argω
N(ω)1−s
∑
τ∈G∗
pℓ
N(τ)≡aN(ω) (mod pℓ)
e
πi
Sp(τ)
pℓ .
Thus, using the estimate of the sum
∑
0 and the Stirling formula for
the gamma-function Γ(z), we at once obtain the estimate of the integral
in (16)
1
2πi
∫ −ε+iT
−ε−iT
Fm(s)
xs
s
ds ≪ T 1+2εp
ℓ
2
+εx−ε ≪ T 1+2εp
ℓ
2 . (17)
Choosing c = 1 + (logx)−1, T = x
1
2
p
3ℓ
4
, we get assertion of Theorem 1.
The following theorems stem from this result and Vinogradov’s lemma
(see, [4], Lemma 12, pp. 261-262).
Theorem 2. In the sectorial region u2 + v2 6 x, u2 + v2 ≡ a (mod pℓ),
ϕ1 < arg (u+ iv) 6 ϕ2, ϕ2 − ϕ1 ≫ x the following asymptotic formula
holds:
A(x;ϕ1, ϕ2; a, p
ℓ) :=
∑
u,v
u2+v2≡a (mod pℓ)
ϕ1<arg (u+iv)6ϕ2
u2+v26x
1 =
=
ϕ2 − ϕ1
2
·
k0x
pℓ
(
1−
χ4(p)
p
)
+O
(
x
1
2
+ε
p
ℓ
4
)
“adm-n2” — 2020/7/8 — 9:42 — page 269 — #129
S. Varbanets, Ya. Vorobyov 269
Theorem 3. Let p be a prime number, ℓ > 3, and p
3ℓ
2−4κ 6 x 6 p2ℓ,
0 < κ 6
1
8 − 1
4ℓ , ϕ2 − ϕ1 ≫ x−κ. Then we have
A(x;ϕ1, ϕ2; a, p
ℓ) =
ϕ2 − ϕ1
2
·
x
pℓ
(
1−
χ4(p)
p
)
+O
(
x1−κ
pℓ
log xκ
)
.
Actually, in Vinogradov’s lemma we take Ω = π
2 , δ = xκ, ∆ = x−α
and let ∆ 6 ϕ2 − ϕ1 <
π
4 − 2κ. Then f(ϕ1, ϕ2) be the function from that
lemma.
Consider the function
Φ(ϕ1, ϕ2) =
1
4
∑
u2+v26x
u2+v2≡a (mod pℓ)
f(arg (u+ iv)).
Then we have
Φ(ϕ1, ϕ2) =
∑
u2+v26x
u2+v2≡a (mod pℓ)
∞
∑
m=−∞
ame4mi arg (u+iv) =
=
∞
∑
m=−∞
am
∑
n≡a (mod pℓ)
n6x
rm(n),
(here am are the coefficients from the Vinogradov’s lemma).
We take r = 3 (in the notation of the Vinogradov’s lemma) and take
into account that
a0 =
1
Ω
(ϕ2 − ϕ1 +∆)
|am| 6
1
Ω(ϕ2 − ϕ1 +∆)
2
π|m| if m 6= 0,
2
π|m|
(
rΩ
π|m|∆
)r
then after simple calculations we get Theorem 2 and Theorem 3.
Taking into account that Hecke characters and Gauss exponential
sums have the multiplicative properties modulo q, we have the following
assertion.
“adm-n2” — 2020/7/8 — 9:42 — page 270 — #130
270 Norm of Gaussian integers
Theorem 4. In the sectorial region u2 + v2 6 x, u2 + v2 ≡ a (mod q),
ϕ1 < arg (u+ iv) 6 ϕ2, ϕ2 − ϕ1 ≫ x the following asymptotic formula
holds:
A(x;ϕ1, ϕ2; a, q) :=
∑
u,v
u2+v2≡a (mod q)
ϕ1<arg (u+iv)6ϕ2
u2+v26x
1 =
=
ϕ2 − ϕ1
2
·
k0x
q
∏
p|q
(
1−
χ4(p)
p
)
+O
(
x
1
2
+ε
q
1
4
)
.
Remark. The result of Theorem 1 can be improved in case p ≡ 3 (mod 4)
and ℓ > 3 in view of the fact that we have the precise meaning of the sum
E0 (see (13)).
References
[1] Iwaniec H., Kowalski E., Analytic Number Theory, Providence: American Mathe-
matical Society. Colloquium Publications, vol. 53, 2004.
[2] Liu K., Shparlinski I.E., Zhang T., Divisor problem in arithmetic progressions
modulo a prime power, Advances in Mathematics, 325(5), 2018, pp. 459-481.
[3] Varbanets S., Exponential sums on the sequences of inversive congruential pseu-
dorandom numbers, Siauliai Math. Semin., 3(11), 2008, pp. 247-261.
[4] Vinogradov I.M., Izbrannye trudy. (Russian) [Selected works.], Izdat. Akad. Nauk
SSSR, Moscow, 1952.
Contact information
Sergey Varbanets Odessa I.I. Mechnikov National University,
Dvoryanskaya str. 2, 65026 Odessa, Ukraine
E-Mail(s): varb@sana.od.ua
Yakov Vorobyov Izmail State Humanities University, Izmail,
Repina str. 12, 68610 Izmail, Ukraine
E-Mail(s): yashavo@mail.ru
Received by the editors: 20.01.2020.
S. Varbanets, Ya. Vorobyov
|
| id | nasplib_isofts_kiev_ua-123456789-188520 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T18:11:32Z |
| publishDate | 2020 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Varbanets, S. Vorobyov, Y. 2023-03-03T19:55:52Z 2023-03-03T19:55:52Z 2020 Norm of Gaussian integers in arithmetical progressions and narrow sectors / S. Varbanets, Y. Vorobyov // Algebra and Discrete Mathematics. — 2020. — Vol. 29, № 2. — С. 259–270. — Бібліогр.: 4 назв. — англ. 1726-3255 DOI:10.12958/adm1529 2010 MSC: 11L07, 11T23 https://nasplib.isofts.kiev.ua/handle/123456789/188520 We proved the equidistribution of the Gaussian integer numbers in narrow sectors of the circle of radius x¹/² , x → ∞, with the norms belonging to arithmetic progression N(α) ≡ ℓ (mod q) with the common difference of an arithmetic progression q, q ≪ x²/³⁻ᵋ. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Norm of Gaussian integers in arithmetical progressions and narrow sectors Article published earlier |
| spellingShingle | Norm of Gaussian integers in arithmetical progressions and narrow sectors Varbanets, S. Vorobyov, Y. |
| title | Norm of Gaussian integers in arithmetical progressions and narrow sectors |
| title_full | Norm of Gaussian integers in arithmetical progressions and narrow sectors |
| title_fullStr | Norm of Gaussian integers in arithmetical progressions and narrow sectors |
| title_full_unstemmed | Norm of Gaussian integers in arithmetical progressions and narrow sectors |
| title_short | Norm of Gaussian integers in arithmetical progressions and narrow sectors |
| title_sort | norm of gaussian integers in arithmetical progressions and narrow sectors |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/188520 |
| work_keys_str_mv | AT varbanetss normofgaussianintegersinarithmeticalprogressionsandnarrowsectors AT vorobyovy normofgaussianintegersinarithmeticalprogressionsandnarrowsectors |